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Transcribed Image Text:5. Now let's think about the bounds of variable . Show that applying the boundary
conditions on forces our quantum number (m) to be equal to 0, ±1, ±2, ±3, ±4 ... What
did applying the boundary condition do to our general solution from question 4?
(Hint: to the apply the boundary condition, ask yourself: how is (p) related to
(+2π)? Once you have the boundary condition, check the appendix for some
potentially useful mathematical expressions)
6. Normalize (d).
7. Prove that a particle-on-a-ring will have discrete energy levels described by the following
equation,
Emi
=
m²ħ²
21
8. Now we will apply these solutions to a Benzene molecule. Assuming benzene is a circle,
estimate the circumference of benzene - assume each carbon-carbon bond in benzene is
1.4 Å. Calculate the radius of benzene using that circumference. There are six л-electrons
in benzene, which are free to move around the ring due to the conjugation of the double
bonds. Make an energy level diagram using the energy expression from question 7, and
calculate the wavelength for the lowest energy electronic transition. The experimentally
observed transition for benzene is at 200 nm, does this agree with your prediction? (Hint:
the ground state will be my = 0, and the rest of the energy levels are going to be doubly
degenerate).
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